Spearing a survivor is the special attack, hitting them, after reeling them in, is just a basic attack. So not even the Yee Haw Shooty Man is an exception.

The Deep Wound is applied when either the Survivor breaks the chain or the Deathslinger lands the basic attack off the chain. If he reels someone in, swings, and misses, no Deep Wound is applied.

I guess the best way to justify it as a basic attack is like this:

Deathslinger's power will make it so when someone is speared by the chain, the next Basic Attack that the Deathslinger performs will inflict the Deep Wound status if it connects with the chained target. If the Survivor breaks the chain, the buff is removed from the Basic Attack and the now-unchained Survivor automatically gains the Deep Wound status.